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In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.

A composition series may not even exist, and when it does, it need not be unique. Nevertheless, a group of results known under the general name Jordan-Hölder theorem asserts that whenever composition series exist, the isomorphism classes of simple pieces (although, perhaps, not their location in the composition series in question) and their multiplicities are uniquely determined. Composition series may thus be used to define invariants of finite groups and Artinian modules.

A related but distinct concept is a chief series: a composition series is a maximal subnormal series, while a chief series is a maximal normal series.

For groups[edit]

Motivation[edit]

If a group G has a normal subgroup N, then the factor group G/N may be formed, and some aspects of the study of the structure of G may be broken down by studying the "smaller" groups G/N and N. If G has no normal subgroup that is different from G and from the trivial group, then G is a simple group. Otherwise, the question naturally arises as to whether G can be reduced to simple "pieces", and if so, are there any unique features of the way this can be done?

Definition[edit]

More formally, a composition series of a group G is a subnormal series of finite length

${\displaystyle 1=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G,}$

with strict inclusions, such that each factor group H_i+1 / H_i is simple.^[1] The factor groups are called composition factors.^[1] Equivalently, a composition series is a subnormal series such that each H_i is a maximal subgroup of H_i+1. Here n is called the length of the series.^[2]

One subnormal series is a refinement of a second subnormal series if every member of the second is a member of the first.^[2] The subnormal series is a proper refinement if the first contains the second properly.^[2] Thus a composition series has no proper refinements; that is, there are no additional normal subgroups which can be "inserted" into a composition series.

If a composition series exists for a group G, then any subnormal series of G can be refined to a composition series.^[3] Every finite group G≠1 has a composition series,^[1] but not every infinite group has one. For example, ${\displaystyle \mathbb {Z} }$ has no composition series.

Uniqueness: Jordan–Hölder theorem[edit]

Statement

Let G be a finite group and let

${\displaystyle 1=N_{0}\leq N_{1}\leq ...\leq N_{r}=G}$

${\displaystyle 1=M_{0}\leq M_{1}\leq ...\leq M_{s}=G}$

be two composition series for G. Then r = s, and there is some permutation p of {1,2,...,r} such that

${\displaystyle M_{p(i)}/M_{p(i)-1}\cong N_{i}/N_{i-1}}$

Proof

By the Schreier refinement theorem, these composition series have equivalent refinements. But by definition these composition series have no proper refinements. Hence the composition series are isomorphic. □

A group may have more than one composition series. However the Jordan–Hölder theorem (named after Camille Jordan and Otto Hölder) shows that any two composition series of a given finite group are in some sense equivalent. The Jordan–Hölder theorem is also true for transfinite ascending composition series, but not transfinite descending composition series^[4].

Example[edit]

The cyclic group C₁₂ has

${\displaystyle C_{1}\triangleleft C_{2}\triangleleft C_{6}\triangleleft C_{12},}$

${\displaystyle C_{1}\triangleleft C_{2}\triangleleft C_{4}\triangleleft C_{12},}$

${\displaystyle C_{1}\triangleleft C_{3}\triangleleft C_{6}\triangleleft C_{12}}$

as different composition series. The sequences of composition factors obtained in the respective cases are

${\displaystyle C_{2},C_{3},C_{2},\ }$

${\displaystyle C_{2},C_{2},C_{3},\ }$

${\displaystyle C_{3},C_{2},C_{2}.\ }$

For modules[edit]

The definition of composition series for modules restricts all attention to submodules, ignoring all additive subgroups that are not submodules. Given a ring R and an R-module M, a composition series for M is a series of submodules

${\displaystyle \{0\}=J_{0}\subset \cdots \subset J_{n}=M}$

where all inclusions are strict and J_k is a maximal submodule of J_k+1 for each k. As for groups, if M has a composition series at all, then any finite strictly increasing series of submodules of M may be refined to a composition series, and any two composition series for M are equivalent. In that case, the (simple) quotient modules J_k+1/J_k are known as the composition factors of M, and the Jordan-Hölder theorem holds, ensuring that the number of occurrences of each isomorphism type of simple R-module as a composition factor does not depend on the choice of composition series.

It is well known^[5] that a module has a finite composition series if and only if it is both an Artinian module and a Noetherian module. If R is an Artinian ring, then every finitely generated R-module is Artinan and Noetherian, and thus has a finite composition series. In particular, for any field K, any finite-dimensional module for a finite-dimensional algebra over K has a composition series, unique up to equivalence.

Generalization[edit]

Groups with a set of operators generalize group actions and ring actions on an group. A unified approach to both groups and modules can be followed as in (Isaacs, Ch. 10) harv error: no target: CITEREFIsaacs (help), simplifying some of the exposition. The group G is viewed as being acted upon by elements (operators) from a set Ω. Attention is restricted entirely to subgroups invariant under the action of elements from Ω, called Ω- subgroups. Thus Ω-composition series must use only Ω subgroups, and Ω-composition factors need only be Ω-simple. The standard results above, such as the Jordan-Hölder theorem, are established with nearly identitical proofs.

The special cases recovered include when Ω=G so that G is acting on itself. An important example of this is when elements of G act by conjugation, so that the set of operators consists of the inner automorphisms. A composition series under this action is exactly a chief series. Module structures are a case of Ω-actions where Ω is a ring and some additional axioms are satisfied.

For objects in an abelian category[edit]

A composition series of an object A in an abelian category is a sequence of subobjects

${\displaystyle A=X_{0}\supsetneq X_{1}\supsetneq \dots \supsetneq X_{n}=0}$

such that each quotient object X_i /X_i + 1 is simple (for 0 ≤ i < n). If A has a composition series, the integer n only depends on A and is called the length of A.^[6]

See also[edit]

• Krohn–Rhodes theory, a semigroup analogue

References[edit]

Notes

Bibliography

• Birkhoff, Garrett (1934), "Transfinite subgroup series", Bulletin of the American Mathematical Society, 40 (12), doi:10.1090/S0002-9904-1934-05982-2
• Isaacs, I. Martin (1994), Algebra: A Graduate Course, Brooks/Cole, ISBN 978-0-534-19002-6
• Kashiwara, Masaki; Schapira, Pierre (2006), Categories and Sheaves
• Dummit, David; Foote, Richard (2004), Abstract Algebra (3 ed.), John Wiley and Sons, Inc., ISBN 0-471-43334-9
• Schenkman, Eugene (1965), Group Theory, Krieger Publishing, ISBN 0-88275-070-4

Category:Subgroup series Category:Module theory